: Since prime numbers are a concept which was invented to be useful for certain
: things, it's important that our choice of including or excluding the number 1
: is constant with our uses for primes. (Just like the definition of 0! is 1,
: because it makes the equations work out, not because of some metaphysical
: argument.)

: It is well known that every counting number ( 0 < n < infinity ; n is integer )
: has a unique prime factorization. If one is a prime number, then the counting
: numbers would all have an infinite number of prime factorizations.

: an example:

: 720 = 2 * 2 * 2 * 2 * 3 * 3 * 5

: If 1 is considered to be a prime number, then the following are all prime
: factorizations also:

: 720 = 1 * 2 * 2 * 2 * 2 * 3 * 3 * 5

: 720 = 1 * 1 * 2 * 2 * 2 * 2 * 3 * 3 * 5

: 720 = 1 * 1 * 1 * 2 * 2 * 2 * 2 * 3 * 3 * 5

: 720 = 1 * 1 * 1 * 1 * 2 * 2 * 2 * 2 * 3 * 3 * 5

: etc.....

: And that's why one isn't a prime number. (among other reasons.)

Not only that but we identify prime number as having only one prime
factor. For 2 it is 2. A composite number is the product of two or
more primes: i.e. 4=2*2. If one were a prime number then numbers like
2,3, 5,7, etc. couldn't be prime. They would be composite. In fact one
would be composite too since it would be 1*1 and would therefore have at
least two factors.